This is a note of real and complex analysis chapter 2.
Chapter 2 is about measures. The measure already defined in chapter 1. In chapter 2, every linear functionals, not combination, of a continuous function space on compact set () () represents the integration of the function () (Riesz representation theorem). Let X be a locally compact Hausdorf space, and let be a positive linear functional on . Then there exist a in which contains all Borel sets in , and there exists a unique positive measure on which represents in the sense that (a) for every and following additional properties:
(b) for every compact set .
(c) For every , we have .
(d) The relation
holds for every open set , and for every with .
(e) If , and , then .
The Riesz theorem is about linear functional is equivalently replaced with choosing measure . Note . The notion of include .
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